Classifying Polygons: Practice

Both opposite-side pairs parallel and all four sides congruent supply a parallelogram with four congruent sides, exactly the rhombus definition used here.

Parallel pairs alone do not force equal sides. A kite pairing alone does not directly supply the stated parallelogram condition.

The least step repeated once per vertex completes a full turn.

A convex regular 20-gon has 20 reflection axes.

The least positive step is 40 degrees.

Four successive matching steps move each vertex four positions forward and match every joining side.

For a half-turn, half of the equal rotation steps must be a whole number of steps.

An odd side count has no integer half. It still has one mirror and one counted rotation per side.

Join two congruent equilateral triangles along a common side, with their apexes on opposite sides. The four outside sides have one common length. The two apex angles are 60 degrees, while each side corner combines two 60-degree angles into 120 degrees.

The line through both apexes reflects the left and right halves onto each other, as in the earlier equilateral-triangle symmetry result. The side and fold conditions hold, but the unequal corner angles prevent regularity.

The square definition includes the parallelogram condition.

Two opposite parallel pairs satisfy at least one opposite parallel pair. No diagonal theorem or orientation argument is needed.

In a convex polygon, a segment between any two points of the region stays inside or on it.

B and D belong to the region, but their joining segment leaves it. Therefore the polygon is not convex; being simple, it is concave.

Fourteen mirrors identify 14 sides for a convex regular polygon. It has 14 rotational symmetries including identity.

Removing identity leaves 13 nonidentity rotations.

The givens establish a rhombus, which is also a kite, parallelogram, and inclusive trapezoid.

Four right angles are not supplied. An oblique rhombus meets every given but is not a square.

When the mirror crosses a side at an interior point, it carries that side onto itself and exchanges its endpoints.

A fixed point on the mirror must be equally far from those exchanged endpoints, so their midpoint is the crossing. This uses the reflection rule, not a future polygon theorem.

For an even side count, opposite sides pair up. Each pair has one midpoint-to-midpoint mirror.

The other eight mirrors pass through opposite vertices.

Four right angles give opposite parallel sides by the earlier perpendicular-line result. With four congruent sides, this gives a square.

Its sides and angles are congruent, making it regular. A convex regular four-gon has four mirrors and four rotations including identity.